This project aims to produce a better definition of the target space for image guided therapy. In order to cover the tumor as it is displaced during treatment, the clinical target must be expanded to cover the region through which it might move. In radiation therapy, the expanded target is known as the Planning Target Volume. The International Commission on Radiation Units and Measurements notes that available planning systems allow only Cartesian expansions that account for the three translational components of motion. The rotational components of displacements are neglected, producing planning target volumes that either miss the true target motion or are larger than needed to cover the target path. As the head rotates in an arc about the long axis of the spine,eg, the planning target volume does not follow the true path of the target that tangents the prevertebral pharyngeal constrictors but juts into these structures to form a box that encloses the curve. The overlap produces higher constrictor doses, a known predictor for late head and neck radiotherapy complications including aspirations which carry mortality risk and necessitate long term feeding tube dependence. Statistical sampling has been examined, mainly in Europe, to draw positions based on observed shifts needed to register planning and treatment images. These methods breakdown because the non commutativity of rotations means that positons can?t be recovered from a uniform sampling of the displacement components. The computational burdens of a six dimensional sampling permuting the rotations is forbidding and rotations away from the target center are neglected This proposal aims to use the projective geometry of displacements expressed as dual quaternions to form the convex hull of the positions swept out by the target as it shifts within its range of translations, rotational angles and rotational axes. Computational experiments will be performed using head and neck cases, to demonstrate generation of planning target volumes within an acceptable time frame (0.25hr). The planning target volume size, coverage, and overlap with the pharyngeal constrictors produced by forming the convex hull of displacements generated by off axis rotations and translations will be compared to simple Cartesian expansions. The long term goal is to reduce treatment morbidity and improve target coverage. The reduction in morbidity, such as aspiration risk, from substituting the new method for target volume creation for conventional simple isotropic expansion is testable in future clinical trials.